Considering that Durmstrang serves as a school for so many different countries we can safely conclude that there will be a variety of different languages.
Beauxbatons Academy has a preponderance of French students, though Spanish, Portuguese, Dutch, Luxembourgians and Belgians also attend in large numbers (both Beauxbatons and Durmstrang have a larger studentship than Hogwarts).
Pottermore, Beauxbatons Academy of Magic
We know it serves many different countries as it is one of three schools in Europe. Hogwarts handles the UK, Beauxbatons Academy of Magic handles predominantly the French, but also accepts the Spanish, Portuguese, Dutch, Luxombourgians and Belgians. If we presume that Durmstrang handles the rest (excluding Russian and possibly some of the Eastern European countries) then that means it can possibly be handling 15 different countries. Although it is possible some of these countries send students to the African school, but that's besides the point.
Students from the Russian Wizarding school, Koldovstoretz, play a version of Quidditch where they fly on entire, uprooted trees instead of broomsticks.
Of the Eleven wizarding schools in the world, the African school of Uagadou is the only one to select pupils by Dream Messenger, leaving a token in the child’s hand whilst they sleep.
http://houseofblackreviews.tumblr.com/post/103497157292/facts-from-the-2014-uk-editions-of-harry-potter
This is information that was published in the 2014 UK editions of Harry Potter
Let's say then we have a low estimate of 10 different languages coming into Durmstrang, now as we know wizards attend muggle school until the age of 10. Schools in Europe can start teaching English from an extremely young age, it's also possible that because of their situation the parents would teach their children English. It's possible that given their location they'd be more likely to learn a more popular language. I am sticking with English due to the evidence we have in the books of all / vast majority of visiting students speaking fluent English to the Hogwarts students.
We know this Tradition of the Tri-Wizard tournament has been going on for centuries enough for Hogwarts to have 63 wins and Beauxbatons to have 62 wins. Which means they have always had to communicate with each other leading me to believe English at least between these three historic schools has been a common language.
Beauxbatons has always enjoyed a cordial relationship with Hogwarts, though there has been a healthy rivalry in international competitions such as the Triwizard Tournament, in which Beauxbatons has sixty-two wins to Hogwarts' sixty-three.
Pottermore, Beauxbatons Academy of Magic
Speculation
Of course, it is unlikely that every student knows English upon arrival, perhaps they have a language year?
Perhaps they have translating magical quills, ROONIL WAZLIB.
Perhaps there is a spell similar to sonorus or even sonorus itself may translate the speech into whoever is hearing it. (This actually leads me onto a slightly less speculative tangent of thought) At the Quidditch World Cup attended by thousands of witches and wizards from all over the world, they would either need magical translation or an understanding of one common language. Of course it considering we see magical glasses which can rewind what you just saw it is not hard to make a jump to a magical hearing aid which translates languages.